lecture note : pramudiyanto, m.eng. -...

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Lecture note :Pramudiyanto, M.Eng.

g{x V|ä|Ä tÇw cÄtÇÇ|Çz XÇz|ÇxxÜ|Çz Xwâvtà|ÉÇ WxÑtÜàÅxÇàFaculty of Engineering, State University of Yogyakarta

Strength of Materials

Pure Bending

04


Pure Bending: Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane


• Principle of Superposition: The normal stress due to pure bending may be combined with the normal stress due to axial loading and shear stress due to shear loading to find the complete state of stress.

• Eccentric Loading: Axial loading which does not pass through section centroid produces internal forces equivalent to an axial force and a couple

• Transverse Loading: Concentrated or distributed transverse load produces internal forces equivalent to a shear force and a couple


MdAyM

dAzMdAF

xz

xy

xx

00

• These requirements may be applied to the sums of the components and moments of the statically indeterminate elementary internal forces.

• Internal forces in any cross section are equivalent to a couple. The moment of the couple is the section bending moment.

• From statics, a couple M consists of two equal and opposite forces.

• The sum of the components of the forces in any direction is zero.

• The moment is the same about any axis perpendicular to the plane of the couple and zero about any axis contained in the plane.


Beam with a plane of symmetry in pure bending:

• member remains symmetric

• bends uniformly to form a circular arc

• cross-sectional plane passes through arc centerand remains planar

• length of top decreases and length of bottom increases

• a neutral surface must exist that is parallel to the upper and lower surfaces and for which the length does not change

• stresses and strains are negative (compressive) above the neutral plane and positive (tension) below it


Consider a beam segment of length L.

After deformation, the length of the neutral surface remains L. At other sections,

mx

mm

x

cy

cρc

yyL

yyLL

yL

or

linearly) ries(strain va


• For a linearly elastic material,

linearly) varies(stressm

mxx

cy

EcyE

• For static equilibrium,

dAyc

dAcydAF

m

mxx

0

0

First moment with respect to neutral plane is zero. Therefore, the neutral surface must pass through the section centroid.

• For static equilibrium,

IMy

cy

SM

IMc

cIdAy

cM

dAcyydAyM

x

mx

m

mm

mx

ngSubstituti

2


• The maximum normal stress due to bending,

modulussection

inertia ofmoment section

cIS

ISM

IMc

m

A beam section with a larger section modulus will have a lower maximum stress

• Consider a rectangular beam cross section,

Ahbhhbh

cIS 6

1361

3121

2

Between two beams with the same cross sectional area, the beam with the greater depth will be more effective in resisting bending.

• Structural steel beams are designed to have a large section modulus.


• Deformation due to bending moment M is quantified by the curvature of the neutral surface

EIM

IMc

EcEccmm

11

• Although cross sectional planes remain planar when subjected to bending moments, in-plane deformations are nonzero,

yy

xzxy

• Expansion above the neutral surface and contraction below it cause an in-plane curvature,

curvature canticlasti 1


A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 GPa and neglecting the effects of fillets, determine (a) the maximum tensile and compressive stresses, (b) the radius of curvature.

SOLUTION:

• Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

2dAIIAAyY x

• Apply the elastic flexural formula to find the maximum tensile and compressive stresses.

IMc

m

• Calculate the curvature

EIM

1


SOLUTION:

Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.

mm 383000

10114 3

AAyY

3

3

3

32

101143000104220120030402109050180090201

mm ,mm ,mm Area,

AyA

Ayy

49-3

2312123

121

231212

m10868 mm10868

18120040301218002090

I

dAbhdAIIx


• Apply the elastic flexural formula to find the maximum tensile and compressive stresses.

49

49

mm10868m038.0mkN 3

mm10868m022.0mkN 3

IcM

IcM

IMc

BB

AA

m

MPa0.76A

MPa3.131B

• Calculate the curvature

49- m10868GPa 165mkN 3

1

EIM

m 7.47

m1095.201 1-3


BENDING OF A MEMBER MADE FROM SEVERAL MATERIAL


• Consider a composite beam formed from two materials with E1 and E2.

• Normal strain varies linearly.

y

x

• Piecewise linear normal stress variation.

yEEyEE xx

222

111

Neutral axis does not pass through section centroid of composite section.

• Elemental forces on the section are

dAyEdAdFdAyEdAdF

222

111

1

2112 E

EndAnyEdAynEdF

• Define a transformed section such that

xx

x

nI

My

21


Bar is made from bonded pieces of steel (Es = 29x106 psi) and brass (Eb = 15x106 psi). Determine the maximum stress in the steel and brass when a moment of 40 kip*in is applied.

SOLUTION:

• Transform the bar to an equivalent cross section made entirely of brass

• Evaluate the cross sectional properties of the transformed section

• Calculate the maximum stress in the transformed section. This is the correct maximum stress for the brass pieces of the bar.

• Determine the maximum stress in the steel portion of the bar by multiplying the maximum stress for the transformed section by the ratio of the moduli of elasticity.


• Evaluate the transformed cross sectional properties

4

31213

121

in063.5

in 3in. 25.2

hbI T

SOLUTION:• Transform the bar to an equivalent cross section

made entirely of brass.

in 25.2in 4.0in 75.0933.1in 4.0

933.1psi1015psi1029

6

6

T

b

s

b

EEn

• Calculate the maximum stresses ksi 85.11

in 5.063in 5.1inkip 40

4

I

Mcm

ksi 85.11933.1max

max

ms

mb

n

ksi 22.9

ksi85.11

max

max

s

b


• Concrete beams subjected to bending moments are reinforced by steel rods.

• In the transformed section, the cross sectional area of the steel, As, is replaced by the equivalent areanAs where n = Es/Ec.

• To determine the location of the neutral axis,

0

022

21

dAnxAnxb

xdAnxbx

ss

s

• The normal stress in the concrete and steel

xsxc

x

nI

My

• The steel rods carry the entire tensile load below the neutral surface. The upper part of the concrete beam carries the compressive load.


A concrete floor slab is reinforced with 5/8-in-diameter steel rods. The modulus of elasticity is 29x106psi for steel and 3.6x106psi for concrete. With an applied bending moment of 40 kip*in for 1-ft width of the slab, determine the maximum stress in the concrete and steel.

SOLUTION:

• Transform to a section made entirely of concrete.

• Evaluate geometric properties of transformed section.

• Calculate the maximum stresses in the concrete and steel.


SOLUTION:• Transform to a section made entirely of concrete.

2285

4

6

6

in95.4in 206.8

06.8psi 106.3psi 1029

s

c

s

nA

EEn

• Evaluate the geometric properties of the transformed section.

422331 in4.44in55.2in95.4in45.1in12

in450.10495.42

12

I

xxxx

• Calculate the maximum stresses.

42

41

in44.4in55.2inkip4006.8

in44.4in1.45inkip40

IMcn

IMc

s

c

ksi306.1c

ksi52.18s


THANK YOU …That’s for now

lecture note : pramudiyanto, m.eng. -...

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